Rational A-functions with rational coefficients

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.
Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.
The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
In our paper we consider the functional system of rational functions with rational coefficients endowed with the superposition operation. We investigate the special case of the completeness problem which is of a particular interest, namely obtaining complete systems of minimum cardinality, i.e. complete systems consisting of a single rational function (such functions are referred to as $A$-functions and are analogues of Schaeffer stroke in Boolean logic).
The main results of the paper are the following:

• there exists an $A$-function;
• the cardinality of the set of all $A$-functions equals $c_{0}$;
• a number of examples of $A$-functions are presented explicitly.